Current measurement systems depend on countries and individuals. There are numerous systems and they are barely harmonized. Nevertheless, the decimal system is now widely used in the world and the metric system created in France in 1795 has attained a certain degree of success during the past two centuries. An international system of units was created in 1960 and international units were defined.
Thus, the international system defined seven base units, i.e., the meter (length), the kilogram (mass), the second (time), the ampere (intensity of the electric current), the Kelvin (temperature), the mole (amount of matter) and the candela (luminous intensity). Among these seven base units, six use the decimal system as multiple or submultiple factors. Thus, for the multiples of these units the terms of deca (10), hecto (102), kilo (103), mega (106), giga (109), etc. are used. For the submultiples use is made symmetrically of the terms deci (10−1), centi (10−2), milli (10−3), micro (10−6), nano (10−9), etc.
However, the decimal system is not used for the measurement of time, but rather a mixed system employing partially the base 60: the unit being the second, 60 seconds forming one minute, 60 minutes forming one hour and 24 hours forming one day, one year being constituted by 365 days.
Neither was the decimal system retained for the measurement of angles: a circle is comprised of 360 degrees (4 times 90 degrees), with the submultiples of the degree being the minute (one degree comprises 60 minutes) and the second (one minute comprises 60 seconds).
It can be useful to question the relevance of the choices that were made when adopting these various units and base units of the international system since, in fact, these units exhibit a certain degree of incoherence: a decimal system was chosen in one case (length), a hexadecimal system was selected in a second case and in a third the system is only partially hexadecimal (90 then 60).
It will furthermore be seen that quite recently (during the 1990s), in the field of geography, still a fourth more complex system was introduced for defining the geographic positioning of a point on Earth: in fact, not two, but three different systems of units are used in GPS devices (“Global Positioning Systems”) which make it possible to specify from satellites the position of a point on the surface of the Earth or in space.
The latitude and longitude of a point are defined in GPS first in terms of degrees (base 360), then in minutes (base 60) and then finally not in seconds (base 60) but in thousandths of minutes (base 1000) for the majority of receivers.
Professional GPS receivers, which are considerably more precise, use the degree, the minute, the second and then decimals of seconds (tenths, hundredths, thousandths, etc.).
It should also be noted that for the measurement of lengths, the meter, the measurement unit of the international system, is not used in numerous fields: thus, in maritime navigation the nautical mile is used (one arc minute measured on the meridian, i.e., 1852 m). In aeronautical navigation, altitudes are measured at the international level in feet (one foot is equal to 33 cm) and not in meters.
Three units have been defined for the measurement of plane angles: the radian, the degree and the grad. The radian is the angle which, having its vertex at the center of a circle, intercepts on the circumference of this circle an arc of a length equal to the radius of this circle. Thus, if the radius of the circle is set to be equal to 1, the radian is the angle which intercepts on the circumference an arc also measuring 1, the perimeter of the circle then being equal to 2 π.
The degree is defined as the angle representing the 360th part of the circle, with the circle being able to be divided into 4 parts equal to 90°. One degree is equivalent to 60 minutes and one minute is equivalent to 60 seconds of arc. The degree is a sexagesimal unit of arc measurement.
The grad is defined as the angle which represents the 400th part of the circle, it being possible to divide the circle into 4 parts equal to 100 grads (or gons). One grad is equivalent to 10 decigrads, 100 centigrads or 1000 milligrads. The grad is a decimal unit of arc measurement.
Thus, we have the following equalities for the measurement of plane angles:
2π radians=360°=400 grads.
1 radian =180/π=57°17′44″=63.662 gons or grads.
It can be seen that the division of the circle into degrees, minutes or seconds is not very practical for the units smaller than the second of an angle, units which are very widely used especially in astronomy. If we followed the base 60 logic that we would have defined a unit which is the 60th part of a second and we would have thus continued the division of the arc units in a sexagesimal system. This did not occur and it seemed to be easier for users to use the decimal system for angle measurement units smaller than the second. Thus, the position of stars in the sky is defined with precision expressed in hundredths, thousandths or ten-thousandths of a second. Thus, telescopes use these units which are more rarely used in nonprofessional astronomical telescopes.
In contrast, it is surprising to note that the use of the grad is extremely common in topography for triangulations and the measurement of angles when performing point surveys. Tachymeters, theodolites, holometric or leveling alidades and angle measurement systems in fact often use either percentages or centesimal minutes as measurement units for slopes or angles. In aerial or spatial photogrammetry the view-finding and photographs are performed using viewfinders often defined first in degrees for the field objectives (90°, . . . ): analogical or analytical stereorestorers used in aerotriangulation, etc.
With regard to the positioning of a point on the surface of the Earth, it has been implemented for many centuries and still at present on the basis of two measurements: its longitude and its latitude, both expressed in degrees. The use of the radian and of the grad is in fact not widely used for defining the coordinates of a terrestrial point.
For longitude, the international reference is the Greenwich meridian: longitude is defined as ranging from 0 to 180° to the east of the Greenwich meridian and from 0 to 180° to the west of the Greenwich meridian. The reference for latitude is the equator: north latitude is defined as 0 to 90° going from the equator to the North Pole and south latitude as 0 to 90° going from the equator to the South Pole. This system is based on the principle of the division of the Earth's surface into four principal zones: the Earth is cut vertically into two parts at the meridian set since 1884 in Greenwich, a town located to the east of London, England. The Earth is also cut into two parts horizontally at the equator. These four zones are represented schematically in FIG. 1.
Any point on the Earth is necessarily situated in one of these four zones and all of the geographic coordinate systems used at present, notably for maritime, aerial and more recently terrestrial navigation, are based on this principle. The GPS satellite positioning referencing system is also based on this principle of dividing the Earth.
Limiting it to whole degrees, this position referencing system divides the Earth into four zones of 16,200 sectors (180×90), i.e., a total of 64,800 sectors (4×16,200). This system has the advantage of being very simple and conforming to the classic position referencing system for points in a plane in which four zones are also distinguished:                zone 1: ++: positive abscissas and ordinates (Northeast)        zone 2: +−: positive abscissas and negative ordinates (Southeast)        zone 3: −+: negative abscissas and positive ordinates (Northwest)        zone 4: −−: negative abscissas and ordinates (Southwest).        
By merging the abscissas with the meridians and the ordinates with the parallels, the axis of the abscissas represents the equator and the axis of the ordinates represents the Greenwich meridian.
However, this position referencing system has the major drawback of using a partially sexagesimal system for angles. This principle uses a first division performed on the basis of a circle of 360 degrees (divisible 4 times 90 degrees, which allows definition of straight angles (180°) and right angles (90°). Then there is a sexagesimal division, passing to a base 60 for the minutes and seconds. Thus, there is neither continuity nor logical integrity between the primary unit (the degree) and the secondary units (the minute which is equivalent to 1/60 of the degree) and the second (which is equivalent to 1/60 minute and 1/3600 degree).
To eliminate this drawback, it has been proposed to use a division of the circle not into degrees, but into grads (a circle of 400 grads is divided into 4 sectors comprised of 100 grads each). This division conserves the straight angles (200 grads) and the right angles (100 grads) and is continuous in its decimals since the secondary units are the decigrad ( 1/10 grad), the centigrad ( 1/100 grad) and the milligrad ( 1/1000 grad). It is this second system which is widely used in France by the National Geographic Institute (IGN), in particular, in the Lambert 2 projections used on all IGN maps.
Whichever of the divisions above is used, this position referencing principle has the disadvantage of referencing a point without taking into account the precision of the measurement nor providing information on this precision. An arc degree, thus, represents along a meridian 111 km, a minute 1,852 m and a second 30.9 m. In reality, the coordinates of a point do not define a point, but rather a zone whose dimensions (expressed in degrees, minutes or seconds) depend on the precision or imprecision of the measurement.
It would accordingly be advantageous to eliminate these drawbacks by providing a localization system for localizing a zone in space in relation to a predetermined point situated on a surface.